The Frostman Shift of an inner function at the value which assumed infinitely and is not an asymptomatic value is an infinite Blaschke Product. But how to charaterize it when it is an asymptotic value?

By Fatou's theorem, the radial limit function
$$\phi^{\ast}(\zeta):=\lim_{r\rightarrow1^{-}}\phi(r\zeta),$$
for a bounded analytic function $\phi$ on $\mathbb{D}$, exists for $m$-almost every $\zeta\in\partial\mathbb{D}$(here $m$ is a normalized Lebesgue measure on $\partial\mathbb{D}$). If $|\phi^{\ast}(\zeta)|=1$ for almost every $\zeta$, then $\phi$ is called an *inner function*.

An inner function can be factored as $$\phi(z)=e^{i\gamma}z^pB(z)\exp(-\int_{\partial\mathbb{D}}\frac{\zeta+z}{\zeta-z}d\mu(\zeta)).$$ Here $\mu$ is a positive finite measure on $\partial\mathbb{D}$ with $\mu\bot m$ and $B(z)$ is a Blaschke Product.

The *Frostman shift*
$$\phi_a(z):=\tau_a\circ\phi(z)=\frac{\phi(z)-a}{1-\overline{a}\phi(z)}, |a|<1$$
are certainly inner functions when $\phi(z)$ is an inner function.

text- the spacing is not quite right, for a start. Please take this as a friendly suggestion for future use of LaTeX – Yemon Choi Aug 24 '13 at 3:54